Data Structures for Moving Objects

Transcription

1 Data Structures for Moving Objects Pankaj K. Agarwal Department of Computer Science Duke University Geometric Data Structures S: Set of geometric objects Points, segments, polygons Ask several queries on S ffl Range searching ffl Nearest-neighbor searching Quad Tree kd Tree BSP Data Structures for Moving Objects 1

2 Moving Objects: Applications Traffic management ffl Location based services ffl Emergency services ffl Air traffic control Digital battlefields Molecular biology Deformable objects Adhoc networks Need data structures for storing, analyzing, querying moving objects. Data Structures for Moving Objects 2 Modeling Motion p(t) =(x(t);y(t)): Position of p at time t. ffl x( );y( ): Polynomials ffl Degree of motion: max degree of x( );y( ). ffl Linear motion: Degree of motion is 1 p(t) =at + b; a; b 2 R 2 p(t) Mostly assume motion to be linear Trajectory of points can change Trajectory can be piecewise linear Issues: Sampled motion Hierarchical motion Uncertainty Data Structures for Moving Objects 3

3 Range Searching S: Set of points Preprocess S into a data structure Report all points of S lying inside a query rectangle Data Structure Space Query Range tree n log n log n + k log log n p kd-tree n n + k External memory data structures also available Example: R-tree Data Structures for Moving Objects 4 Kinetic Range Searching S: Set of points, each moving with fixed velocity in the plane Preprocess S into a data structure: Q1 Given a rectangle R and a time value t, report all points of S(t) R Q2 Given a rectangle R and time values t 1 ;t 2, report all points that pass through R during the time interval [t 1 ;t 2 ]. t t y y x x Data Structures for Moving Objects 5

4 Early Approaches One-dimensional data structures [Wolfson et al , Tayeb et al. 98, Kollios et al. 99] Two-dimensional data structures ffl Map trajectories to higher dimensional points [Kollios et al.] ffl Build index on trajectories [Pfoser et al.] ffl Parametric R-trees [Saltenis et al.] Assumes frequent updates on trajectories Data Structures for Moving Objects 6 Kinetic Range Searching (A., Arge, Erickson, 2001) Partition-tree based approach ffl O(n) space, s p n + k query time ffl log 2 n insertion/deletion/trajectory-change ffl oblivious scheme Kinetic range trees ffl n log n= log log n space, log n + k query ffl Events: x- ory-coordinates of two points become equal ffl (n 2 ) events, each requiring log 2 n time ffl Tradeoff between # events and query time ffl Queries have to arrive in a chronological order Data Structures for Moving Objects 7

5 Partition Tree Based Approach t t xt* yt* t y x y x Trajectory of a point p i is a line `i in R 3 p i (t) 2 R () `i intersects (R; t) `x;`y: Projection of ` onto the xt- &yt-planes ` intersects (R; t), `x intersects (R x ;t) & `y intersects (R y ;t) Use duality and partition trees Data Structures for Moving Objects 8 R-Trees Bounding box hierarchy, B-tree Each node v is associated with a subset S v of points and the smallest rectangle R v containing S v Partition S v into B clusters, each associated with a child of v Several heuristics are proposed for partitioning S v into B clusters F G R1 R2 R1 B E H R5 R3 R4 R5 R6 A R3 C I R2 A B C D E F G H I R4 D Data Structures for Moving Objects 9

6 STAR-Tree Kinetic R-tree Maintain the smallest box enclosing the set of moving points ffl Box is defined by four points ffl The combinatorial structure can change Ω(n) times B(t) ffl Maintain an approximation of the smallest enclosing box Maintaining the clustering kinetically ffl Extend the known heuristics ffl No theoretical nontrivial results known on kinetic clustering Data Structures for Moving Objects 10 Smallest Enclosing Box R(P (t)): Smallest box enclosing P at time t "-core-set: C S"-coreset if 8t (1 ")R(S(t)) R(C(t)) Theorem: 9 "-core-set of size 1= p "; Computation time: n +1=" A more general result on core sets in [A., Har-Peled, Varadarajan] Leads to approximatation algorithms for several problems Data Structures for Moving Objects 11

7 Smallest Enclosing Box 1 1 Quality Kernel Size = 16 Kernel Size = Quality Kernel Size = 16 Kernel Size = Linear Motion, Input Size = 10,000 Quadratic Motion, Input Size = 10, Linear Motion, Kernel Size = 24 Quadratic Motion, Kernel Size = # Events # Events Data Structures for Moving Objects 12 Smallest Enclosing Box STAR-tree: Maintain a box enclosing S v at each node v Compute C v ρ S v for each node in a bottom-up manner ffl Merge the core sets computed at the children of v ffl Prune the merged set Maintain the smallest enclosing box R(C v ) Data Structures for Moving Objects 13

8 Re-Clustering Reorganize the children of a node if the rectangles of their children overlap a lot. u 2 u 3 u 3 w 2 u 1 v w 1 v Collect all the grandchildren of the node Reconstruct a 2-level R-tree on them Data Structures for Moving Objects 14 Experimental Results Synthetic Data 100, ,000 points inside km 2 area with different distributions Points are inserted/deleted dynamically, at any time at least 80% points present Three range of speed: 45 km/h, 75km/h, 180 km/h Search I/O S1 (approx.) S1 (exact) S2 (approx.) S2 (exact) S3 (approx.) S3 (exact) Search I/O Number of points (x 1000) Data Structures for Moving Objects 15

9 Experimental Results Realistic Data Extracted the roads map around Durham, NC, within 120 miles centered at Durham (ß 250; 000 polygonal chains) Computed a planar map of the road network Chose source and destinations randomly with some distribution Computed a good path using Dijkstra s algorithm minimize the length + number of turns Used Douglas Peucker algorithm to simplify the paths Data Structures for Moving Objects 16 Experimental Results Avg. Query I/O Data Structures for Moving Objects 17

10 Tradeoffs in Performance Accuracy vs efficiency ffl Maintain approximate structures Query vs events ffl Combine KDS and time-oblivious approaches Responsive Approach: ffl Near-future queries are more critical than far-future queries ffl Fast query time for near-future queries ffl Measure future by the number of events occurred ffl # events:, query: p =n + k Data Structures for Moving Objects 18 Concluding Remarks Incorporating more realistic motions ffl Use dynamic systems, e.g., Kalman, particle filters, to model trajectories ffl How does one perform geometric computation in this model? ffl Geometric computation under uncertainty Hierarchical representation of motion Kinetic data structure for clustering, similarity searching Data Structures for Moving Objects 19